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Solutions of 3D Reflection Equation from Quantum Cluster Algebra Associated with Symmetric Butterfly Quiver

Rei Inoue, Atsuo Kuniba

TL;DR

This work constructs a novel pair of operators $(R,K)$ that solve the three-dimensional reflection equation by embedding the problem in a quantum cluster-algebra framework based on Symmetric Butterfly quivers of type $C$. The $R$-operator matches the previously known construction built from four quantum dilogarithms, while the new $K$-operator involves ten dilogarithms, with both decomposed into a dilogarithm part and a monomial part that can be moved by adjoint actions. The core technical achievement is a full 3DRE verification: under carefully chosen parameter constraints, the $R$- and $K$-operators satisfy the boundary analogue of the tetrahedron equation, with a 46-term dilogarithm identity and a well-defined algebraic structure in both quantum-$Y$ and $q$-Weyl representations. The paper also shows how the SB-construction reduces to the FG quiver setting, linking the new boundary solution to established 2D boundary integrable structures and suggesting routes to broader applications and matrix-element calculations. Overall, the work deepens the connection between quantum cluster algebras, dilogarithm identities, and higher-dimensional integrable boundary systems, offering a robust framework for future boundary solvable models.

Abstract

We construct a new solution $(R,K)$ to the three-dimensional reflection equation, a boundary analogue of the tetrahedron equation. The $R$-operator is the one obtained by Sun, Terashima, Yagi, and the authors in 2024, involving four quantum dilogarithms with arguments in the $q$-Weyl algebra. The new $K$-operator similarly involves ten such quantum dilogarithms. Our approach is based on the quantum cluster algebra associated with the symmetric butterfly quiver on the wiring diagram of type C.

Solutions of 3D Reflection Equation from Quantum Cluster Algebra Associated with Symmetric Butterfly Quiver

TL;DR

This work constructs a novel pair of operators that solve the three-dimensional reflection equation by embedding the problem in a quantum cluster-algebra framework based on Symmetric Butterfly quivers of type . The -operator matches the previously known construction built from four quantum dilogarithms, while the new -operator involves ten dilogarithms, with both decomposed into a dilogarithm part and a monomial part that can be moved by adjoint actions. The core technical achievement is a full 3DRE verification: under carefully chosen parameter constraints, the - and -operators satisfy the boundary analogue of the tetrahedron equation, with a 46-term dilogarithm identity and a well-defined algebraic structure in both quantum- and -Weyl representations. The paper also shows how the SB-construction reduces to the FG quiver setting, linking the new boundary solution to established 2D boundary integrable structures and suggesting routes to broader applications and matrix-element calculations. Overall, the work deepens the connection between quantum cluster algebras, dilogarithm identities, and higher-dimensional integrable boundary systems, offering a robust framework for future boundary solvable models.

Abstract

We construct a new solution to the three-dimensional reflection equation, a boundary analogue of the tetrahedron equation. The -operator is the one obtained by Sun, Terashima, Yagi, and the authors in 2024, involving four quantum dilogarithms with arguments in the -Weyl algebra. The new -operator similarly involves ten such quantum dilogarithms. Our approach is based on the quantum cluster algebra associated with the symmetric butterfly quiver on the wiring diagram of type C.
Paper Structure (43 sections, 27 theorems, 210 equations, 7 figures)

This paper contains 43 sections, 27 theorems, 210 equations, 7 figures.

Table of Contents

  1. Introduction and main result
  2. Background
  3. New solution to 3DRE
  4. Method
  5. Outlook
  6. Outline of the paper
  7. Basics of quantum cluster mutations and notations
  8. Quantum cluster mutation
  9. Periodicity and quantum dilogarithm identity
  10. $q$-Weyl algebras
  11. Other notations
  12. Tetrahedron and 3D reflection equations from cluster mutations
  13. $R$-operator
  14. $K$-operator
  15. 3D reflection equation
  16. ...and 28 more sections

Key Result

Theorem 2.1

For an exchange matrix $B$ and a mutation sequence $\nu$ for $B$, the following two statements are equivalent:

Figures (7)

  • Figure 3.1: The tetrahedron relation for SB.
  • Figure 3.2: The mutation sequence for $\mathcal{K}_{1234}$.
  • Figure 3.3: SB quivers connected by the two sides of the 3D reflection equation.
  • Figure 3.4: The LHS of the 3D reflection equation \ref{['RE-SB']}. Blue vertices have weight two.
  • Figure 3.5: The RHS of the 3D reflection equation \ref{['RE-SB']}. Blue vertices have weight two.
  • ...and 2 more figures

Theorems & Definitions (43)

  • Theorem 2.1
  • Lemma 2.2
  • proof
  • Remark 3.1
  • Lemma 3.2
  • Proposition 3.3
  • proof
  • Proposition 4.1
  • proof
  • Remark 4.2
  • ...and 33 more